1. Field of the Invention
The present invention relates to a method of designing digital filters, and more particularly, to a method of designing low-order linear-phase IIR filters for approximating an FIR filter.
2. Description of Related Art
Digital filters for addressing digital signal processing have been applied widely in commercial electronic products, such as compact disk players, television sets and the like. In order to deal with real-time signals, the designs should be considered low computational costs to output signals efficiently. Also, since the distortion-free transmission of-the waveforms in the passband is very important in signal processing, the filters should contain a linear-phase characteristic, i.e., constant group delay.
There are two classes of digital filters. One is a finite-duration impulse response (FIR) filters and the other is an infinite-duration impulse response (IIR) filters. The main advantages of the FIR filters are exactly linear phase and guaranteed stable. However, when the specification being very rigorous, the resulting FIR filter is usually with higher orders, which may require more hardware components and lower the operational speed. Conversely, the IIR filters are useful for large-scale or high-speed designs but they do not have exactly linear phase and can not guarantee to be stable.
One way to synthesize the IIR filters with linear phase in passband is to solve the rational approximation problem directly. These methods, for example, include Pade approximation, linear programming, nonlinear programming, multiple criterion optimization and eigenfilter approach.
Another way is called the indirect approach. It will be composed of three steps, as shown in FIG. 1. Step 1 receives and stores the design specifications of a digital filter.
These specifications are required in the frequency-domain in terms of the desired magnitude and phase response of the filter. Then, a linear-phase FIR filter, which meets design specifications, will be designed in step 2. The order and the coefficients of the FIR filter can be obtained using the conventional methods such as the frequency-sampling design technique, the window design technique and the optimal equiripple design technique. Finally, a lower-order IIR filter will be obtained using filter approximation techniques in step 3. It should be mentioned that the special attentions shall be paid on this indirect approach. The resulting IIR filter must be ensured to capture the linear-phase response of the original FIR filter in the passband.
In recent years, several linear-phase IIR filter design techniques have been emerged for this purpose. Generally speaking, two distinct methods have be proposed: (1) Grammian-Based Methods: including the balanced truncation method and impulse response grammian method and (2) Optimal Approximation Methods: including the least-square approximations and the H2 norm approximation. Although satisfactory results have been reported, computational complexity of these methods are still quite expensive.